Acoustics & Music Theory
why it sounds like that

Why it sounds like that.

Music theory is usually taught as a set of rules and acoustics as a set of equations, and the two are the same subject seen from opposite ends. These are essays about the middle — one idea at a time, illustrated until the argument is visible, and where a picture makes a claim about a sound, the sound is there to be played.

Twelve fifths do not make seven octavesPitch drawn as a spiral: one turn is one octave, so a note's angle is its pitch class and its radius is how high it has climbed. Twelve pure fifths wind round just over seven times and finish 23.46 cents past seven octaves — the Pythagorean comma, and the reason no keyboard can be tuned in pure fifths.CC♯DE♭EFF♯GA♭AB♭Bstartseven octavestwelve fifthsthey miss by 23.46 centsone turn is one octave · angle is pitch class · radius is how far it has climbed
Fig. 1 Twelve perfect fifths stacked on top of each other, drawn as a spiral so the ends can be compared. They should arrive back where they started after seven octaves and they miss, by a little under a quarter of a semitone. Every tuning system ever devised is a decision about where to put that gap — and the buttons below the picture play the two ends, which is the fastest way to hear that this is a real discrepancy and not a rounding error.

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19 essays

CC♯DE♭EFF♯GA♭AB♭Bstartseven octavestwelve fifthsthey miss by 23.46 centsone turn is one octave · angle is pitch class · radius is how far it has climbed Pitch and tuning

Twelve fifths and seven octaves, which are not the same thing

Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that.

10 figures 024681012-20-1001020semitones above the tonicCC♯DE♭EFF♯GA♭AB♭BCjust intonationPythagoreanquarter-comma meantonesharp of equalflat of equal Pitch and tuning

Where to hide the comma, which is the only real question

Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written.

9 figures E♭B♭FCGDAEBF♯C♯G♯D♯-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4+35.7each fifth's departure from a pure 3:2, in centsone fifth carries the whole error Pitch and tuning

The wolf at the end of the chain

Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose.

9 figures one frame = one repeat of the combined wavelower — 2 per frameupper — 3 per frametheir sumthe pattern repeats after 2 cycles of the lower tone Intervals and chords

Two notes and a ratio, which is the whole of consonance

Sound two tones together and the pair either settles or does not. What decides it is the ratio of their frequencies, and the rule is that simpler ratios settle — which is two and a half thousand years old and still not quite an explanation.

9 figures 00.511.52seconds3 beats per second — the difference, exactlythe carrier is drawn slower than it sounds, or it would be a solid band Intervals and chords

Beats are arithmetic that anybody can hear

Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned.

9 figures semitones above the lower tone6/55/44/33/25/32/1CC♯DE♭EFF♯GA♭AB♭BCroughest at about a semitonesmooth at the simple ratios Intervals and chords

Roughness can be computed, and the answer looks like a scale

Feed two spectra into a model of how nearby frequencies interfere, sweep one past the other, and the curve that comes out has dips exactly where the consonant intervals are. Nobody put them there.

8 figures CC♯DE♭EFF♯GA♭AB♭B2 · 2 · 1 · 2 · 2 · 2 · 1steps, in semitonesthe short chords are the semitones — two of them, unevenly spaced Scales and modes

Seven of the twelve, chosen unevenly

A major scale is a selection of seven positions out of twelve, and the selection is lopsided on purpose. The two semitones sit where they do because an even choice would destroy the thing that makes a scale usable.

8 figures CC♯DE♭EFF♯GA♭AB♭BCLydianIonianMixolydianDorianAeolianPhrygianLocrianbrightest at the top, darkest at the bottomeach row drops exactly one note by a semitone from the row above Scales and modes

The same seven, started later

A mode is not a new scale. It is the same seven notes with a different one treated as home, and the entire change in character comes from reassigning which degree the semitones fall next to.

8 figures CaGeDbAf♯Ec♯Ba♭F♯e♭C♯b♭A♭fE♭cB♭gFdone step= one fifth= one sharpouter ring: major keys · inner ring: their relative minors Scales and modes

Keys are neighbours, and the map is computed

Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away.

7 figures C1/1E5/4G3/243major0–4–7C1/1E♭6/5G3/234minor0–3–7C1/1E♭6/5F♯33diminished0–3–6C1/1E5/4A♭8/544augmented0–4–8semitones above the root Harmony and voice leading

Three notes at once, and why these three

A triad is two thirds stacked, and the four ways of stacking them behave completely differently. The difference is entirely in whether the resulting ratios are simple.

8 figures 2C majorA minortotal 2 semitones12C majorF majortotal 3 semitones231C majorF♯ majortotal 6 semitonesthe shortest change is the one nobody has to sing far Harmony and voice leading

The shortest move, which is what a chord change is

Between any two chords there is an assignment of voices that moves the least. Compute it and the changes that composers actually use turn out to be the short ones.

8 figures ICiiDiiiEIVFVGviAvii°BI – vi – IV – Vradius = semitones of voice motion from the tonic chord Harmony and voice leading

A progression is a path, and the map can be drawn

Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.

8 figures 12345678x..x..x.3 onsets in 8 stepsthe polygon is what stays the same when the rhythm is rotated Rhythm and metre

Rhythm is a circle, and the bar line is a choice

Draw a rhythm as a line and it looks like a sequence of decisions. Draw it as a cycle and the same pattern turns out to be shared across continents, differing only in where somebody decided to start counting.

7 figures E(2,4)the simplest divisionE(3,8)tresillo — Cuba, and half the world's pop musicE(5,8)cinquilloE(4,9)Turkish aksakE(5,12)West African bell patternE(7,16)Brazilian necklaceone cell per step · filled cells are struck Rhythm and metre

As evenly as possible, which turns out to be a famous rhythm

Ask an algorithm to space five strikes over eight beats as evenly as it can. It produces the cinquillo. Ask for three over eight and it produces the tresillo. Nobody told it about Cuba.

7 figures 32they coincide every 6 subdivisionscommon grid: 6 equal steps Rhythm and metre

Two clocks at once, and where they agree

Three against two repeats every six subdivisions and feels like a figure. Seven against five repeats every thirty-five and feels like weather. The difference is one number.

10 figures 1130.8 HzC2261.6 HzC3392.4 HzG4523.3 HzC5654.1 HzE -14¢6784.9 HzG7915.7 HzB♭ -31¢81046.5 HzCpartialthe dots are the nodes — the places that do not move Timbre and acoustics

A string does everything at once

A plucked string does not vibrate at one frequency. It vibrates at all the whole-number multiples of one frequency simultaneously, and nearly everything in music theory is downstream of that fact.

7 figures 1pureone partial, nothing else12345678stringall partials, falling12345678clarineteven partials nearly absent123456789bellodd partials onlyamplitude Timbre and acoustics

The ear hears the list, not the shape

Two sounds with the same partials and different phases have completely different waveforms and sound identical. What the ear extracts is a list of frequencies and strengths, and everything else is discarded.

7 figures 00.511.5200.20.40.60.81secondsamplitudeplucked — no sustainbowedstruck — no sustainkey released Timbre and acoustics

The shape of a note, which is most of what an instrument is

Cut the first fifty milliseconds off a recorded piano and listeners stop calling it a piano. The attack carries more identity than the steady tone it leads into, and it is the part every spectrum plot leaves out.

6 figures mode 1 · 40.8 Hzloudest at both endsmode 2 · 81.7 Hz2 quiet linesmode 3 · 122.5 Hz3 quiet lines4.2 × 3.4 × 2.5 m — every mode below 160 Hz:4150658094107118129140151gaps here are notes the room does not support Timbre and acoustics

The room is part of the instrument

A room has frequencies it supports and frequencies it will not. In a small one those frequencies are far apart, so some bass notes are loud in one corner and absent in another — and no equipment fixes it.

7 figures

Threads running through

themes, not chapters

Small whole numbers

Consonance is frequency ratios of small integers, and it has been known since Pythagoras. Almost everything interesting follows from those integers refusing to line up.

12 essays

The comma that will not close

Twelve fifths do not equal seven octaves. Every tuning system in history is a different decision about where to hide the difference.

5 essays

The ear is not a microphone

Loudness is not amplitude, pitch is not frequency, and a note whose fundamental is missing still has one. Perception is a model, not a measurement.

7 essays

The same shape, moved

A chord, a scale and a rhythm are all patterns that keep their identity when shifted. What survives the shift is what the theory is about.

6 essays

What the room does

No sound reaches an ear unmodified. Reflection, absorption and resonance are part of the instrument, and always have been.

1 essays

Notation hides things

Staff notation is a five-hundred-year-old compression. What it records beautifully and what it cannot record at all are both worth knowing.

4 essays

Time is a circle

Rhythm drawn as a line looks arbitrary. Drawn as a cycle, the patterns that recur across continents turn out to be the same few shapes.

3 essays